\section{Simple problems}


% Sherlock, A little exercise
\problem{A little exercise}
\difficulty{1}{5}

Black has just moved in the game below. White started on the south side of the board.\par
What was Black's last move, and what was White's last move? \par
\note[Note]{
	The boards below are identical copies. Scribble to your heart's content.\\
	There a few empty boards at the end of this handout as well.
}

% spell:off
\manyboards{
	ka8,Kc8,
	Ph2,
	Bg1
}
% spell:on

\begin{solution}
	It's pretty clear that Black just moved out of check from A7.

	\vspace{2mm}

	How did White deliver this check? The bishop couldn't have moved to G1,
	so this check must have been discovered by another piece. Since there are
	no extra pieces on the board, Black must've captured this piece on his last move.

	\vspace{2mm}

	The only piece that could have moved from the white bishop's diagonal to
	then be captured on A8 is a knight.

	\vspace{2mm}

	\textbf{Note:}
	There are two possible answers if we don't know who started where.
	If Black had started on the south side of the board, that bishop could be a promoted pawn.
\end{solution}

\vfill
\pagebreak










% Sherlock, Which color?
\problem{Which color?}
\difficulty{2}{5}

In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
There is a pawn at G3. What color is it? \par
As before, White started on the bottom of the board.

% spell:off
\manyboards{
	ke8,
	Kb4,
	Ug3,
	Pd2,Pf2
}
% spell:on

\begin{solution}
	The white king is the key to this solution. How did it get off of E1? \par
	It must have castled kingside---castling queenside would move a rook from black to white.

	\vspace{2mm}

	Now, the white king is on G1. How did it get out of there? \par
	It's must have moved through H2 and G3, which would be impossible if the mystery pawn on G3 was white.
	Therefore, that pawn must be black.
\end{solution}

\vfill
\pagebreak













% Arabian Knights 2
\problem{Invisible, but not invincible}
\difficulty{2}{5}

The black king has turned himself invisible. Unfortunately, his position is hopeless. \par
Mate the king in one move. \par

% spell:off
\manyboards{
	Ra8,rb8,Kf8,
	Nb7,Pc7,
	Pa6,Rc6
}
% spell:on

\begin{solution}
	Since it is White's move, Black cannot be in check. \par
	So, either White is in check or the black king is on C8. \par
	If White is in check, Black must have administered this check by moving from C8 to D7. \par
	Therefore, the black king must be on C8 or D7.

	\vspace{2mm}

	If we capture the black rook on B8 with the pawn on C7 and promote it to a knight, the black king will be in checkmate
	regardless of his position.
\end{solution}

\vfill
\pagebreak






% Sherlock, a question of survival
\problem{An empty board}
\difficulty{2}{5}

In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
There is one more piece on the board, which isn't shown. What color square does it stand on? \par

% spell:off
\manyboards{
	ke8,
	Pd2,Pf2,
	Ke1
}
% spell:on

\begin{solution}

	Which piece performed the last capture on a black square? It couldn't have been a white pawn, which haven't moved.
	It couldn't have been the white king, which is trapped; or the black king, which is restricted to white squares.

	\vspace{2mm}

	It must have been the piece we can't see, which therefore stands on a black square.

\end{solution}

\vfill
\pagebreak


% Sherlock, another monochromatic
\problem{The knight's grave}
\difficulty{3}{5}

In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
The white king has made less than fourteen moves. \par
Use this information to show that a pawn was promoted. \par

% spell:off
\manyboards{
	ke8,
	Pb2,Pd2,
	Ke1
}
% spell:on

\begin{solution}
	Knights always move to a different colored square, so all four missing knights must have been captured on their home square.
	What pieces captured them?

	\vspace{2mm}

	We can easily account for the white knights and the black knight on G8, but who could've captured the knight from B8?
	The only white pieces that can move to black squares are pawns, the Bishop (which is trapped on C1), the rook (which is stuck on column A and row 1), or the king (which would need at least 14 moves to do so).

	\vspace{2mm}

	If this knight was captured by a pawn, that pawn would be immediately promoted. If it was captured by a piece that wasn't a pawn, that piece must be a promoted pawn.
\end{solution}

\vfill
\pagebreak






% Arabian Knights, intro (given with solution)
\problem{Promotion?}
\difficulty{2}{5}


It is White's move. Have there been any promotions this game? \par

% spell:off
\manyboards{
	Pb2,Pe2,kf2,Pg2,Ph2,
	Bc1,Kd1,Rh1
}
% spell:on

\begin{solution}

	Since it is White's move, Black has just moved his king. Where did he move it from?
	Not E1, E3, F3, or G3, since that implies Black had moved into check before. \par

	\vspace{2mm}

	The only remaining possibilities are F1 and G1. \par
	G1 is again impossible: how would the king get there without moving into check? \par
	F1, therefore, is the only choice. If we place the king on F1, we see that another piece must prevent check from the white rook.
	This must have been a white black-square bishop, which moved to F2 to reveal that check, and was then captured by the black king.

	\vspace{2mm}

	However, there is already a white black-square bishop on the board! We can get a second only by promoting a pawn, so the answer is \say{yes.}

\end{solution}

\vfill
\pagebreak



% Sherlock Holmes, two bagatelles (1)
\problem{Whodunit}
\difficulty{2}{5}

It is Black's move. Can Black castle? \par
\hint{Remember the rules of chess: you may not castle if you've moved your rook.}

% spell:off
\manyboards{
	ra8,bc8,ke8,rh8,
	pa7,pc7,pe7,pg7,
	pb6,pf6,ph6,
	Pa3,
	Pb2,Pc2,Pd2,Pe2,Pf2,Pg2,Ph2,
	Bc1,Qd1,Ke1,Bf1
}
% spell:on

\begin{solution}
	White's last move was with the pawn. \par
	Black's last move must have been to capture the white piece which moved before that.

	\vspace{2mm}

	This piece would have to have been a knight, since the white rooks could not have got out onto the board.
	It is clear that none of the black pawns captured this knight.
	The black rook on A8 couldn't have captured it either, because there is no square that
	the knight could have moved from to get to that position.

	\vspace{2mm}

	The black bishop couldn't have captured the knight either, since the only square the
	knight could have come from is D6, where it would have been checking the king.

	\vspace{2mm}

	So, the black king or the rook on H8 made this capture. Therefore, Black can't castle.
\end{solution}

\vfill
\pagebreak










% Sherlock Holmes, two bagatelles (2)
\problem{Castle contradiction}
\difficulty{2}{5}

Neither Black nor White captured a piece on their last move. \par
It is Black's move. Can he castle? \par
\hint{What was White's last move? Check the cases.}

\manyboards{
	ke8,rh8,
	pc4,
	Pf3,
	Pc2,Pf2,Pg2,
	bd1,Rf1,Kg1
}


\begin{solution}
	If White's last move was with the king, then the black rook moved to check him and Black can't castle.

	\vspace{2mm}

	If White's last move wasn't with the king, White must have castled. \par
	What was Black's last move? \par
	If it was with the king or rook, Black can't castle.

	\vspace{2mm}

	It could not have been with the bishop, since then White would have had no move immediately before that.
	Now, suppose Black moved his pawn. Then White's preceding move must have been with the pawn from E2,
	capturing a piece on F3. This means that the bishop on D1 is a promoted bishop. The promoting pawn must
	have come from D7, passed D2, checked the white king, making it move!
	This contradicts our assumption that White has just castled.
\end{solution}

\vfill
\pagebreak












% Arabian Knights, intro (given with solution)
\problem{A matter of order}
\difficulty{3}{5}

A black bishop captured a White piece earlier in this game. \par
Which bishop was it, and what did it capture? \par
\hint{Black and White start with 16 pieces each.}

\manyboards{
	ra8,qd8,ke8,
	pa7,pc7,pd7,pf7,ph7,
	pb6,nc6,pe6,nf6,ph6,
	Bb5,be5,
	Pe4,bg4,
	Pc3,Nf3,
	Pa2,Pb2,Pc2,Qe2,Pf2,Pg2,Ph2,
	Kc1,Rd1,Rh1
}

\begin{solution}
	First, notice that the pawn on C3 came from D2 by capturing a piece. \par
	This must have been a black rook, which is the only missing black piece.

	\vspace{2mm}

	This black rook couldn't have moved there before the black pawn on G7 captured a white piece on H6.
	This piece couldn't have been the missing white bishop, because that bishop would still be trapped by the pawn on D2.
	Therefore, the missing white knight was captured on H6.

	\vspace{2mm}

	The only other missing white piece is the black-square bishop, which must have been captured by the black bishop on E5.

\end{solution}

\vfill
\pagebreak